In statistics and probability theory, the median is the number separating the higher half of a data sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one (e.g., the median of {3, 3, 5, 9, 11} is 5). If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values [1][2] (the median of {3, 5, 7, 9} is (5 + 7) / 2 = 6), which corresponds to interpreting the median as the fully trimmedmid-range. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data is contaminated, the median will not give an arbitrarily large result. A median is only defined on ordered one-dimensional data, and is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions.

In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median (in the case of an even sample size); if there is such a member, there may be more than one so that the median may not uniquely identify a sample member. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid. At most, half the population have values strictly less than the median, and, at most, half have values strictly greater than the median. If each group contains less than half the population, then some of the population is exactly equal to the median. For example, if a < b < c, then the median of the list {a, b, c} is b, and, if a < b < c < d, then the median of the list {a, b, c, d} is the mean of b and c; i.e., it is (b + c)/2.

The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors.

In terms of notation, some authors represent the median of a variable x either as or as [1] sometimes also M.[3] There is no widely accepted standard notation for the median,[4] so the use of these or other symbols for the median needs to be explicitly defined when they are introduced.

The median is one of a number of ways of summarising the typical values associated with members of a statistical population; thus, it is a possible location parameter. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles.

For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient than the sample median when data are uncontaminated by data from heavy-tailed distributions or from mixtures of distributions, but less efficient otherwise, and that the efficiency of the sample median is higher than that for a wide range of distributions. More specifically, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean—see Efficiency (statistics)#Asymptotic efficiency and references therein.

Any probability distribution on R has at least one median, but there may be more than one median. Where exactly one median exists, statisticians speak of "the median" correctly; even when the median is not unique, some statisticians speak of "the median" informally.

The median is used primarily for skewed distributions, which it summarizes differently from the arithmetic mean. Consider the multiset { 1, 2, 2, 2, 3, 14 }. The median is 2 in this case, (as is the mode), and it might be seen as a better indication of central tendency (less susceptible to the exceptionally large value in data) than the arithmetic mean of 4.

The mean absolute error of a real variable c with respect to the random variableX is

Provided that the probability distribution of X is such that the above expectation exists, then m is a median of X if and only if m is a minimizer of the mean absolute error with respect to X.[6] In particular, m is a sample median if and only if m minimizes the arithmetic mean of the absolute deviations.

The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes the absolute deviation function

This proof can easily be generalized to obtain a multivariate version of the inequality,[9] as follows:

In individual series (if number of observation is very low) first one must arrange all the observations in order. Then count(n) is the total number of observation in given data.

If n is odd then Median (M) = value of ((n + 1)/2)th item term.

If n is even then Median (M) = value of [((n)/2)th item term + ((n)/2 + 1)th item term ]/2

For an odd number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7.

Start by sorting the values: 1, 2, 5, 7, 8.

In this case, the median is 5 since it is the middle observation in the ordered list.

The median is the ((n + 1)/2)th item, where n is the number of values. For example, for the list {1, 2, 5, 7, 8}, we have n = 5, so the median is the ((5 + 1)/2)th item.

median = (6/2)th item

median = 3rd item

median = 5

For an even number of values

As an example, we will calculate the sample median for the following set of observations: 1, 6, 2, 8, 7, 2.

Start by sorting the values: 1, 2, 2, 6, 7, 8.

In this case, the arithmetic mean of the two middlemost terms is (2 + 6)/2 = 4. Therefore, the median is 4 since it is the arithmetic mean of the middle observations in the ordered list.

We also use this formula MEDIAN = {(n + 1 )/2}th item . n = number of values

As above example 1, 2, 2, 6, 7, 8 n = 6 Median = {(6 + 1)/2}th item = 3.5th item. In this case, the median is average of the 3rd number and the next one (the fourth number). The median is (2 + 6)/2 which is 4.

The distribution of both the sample mean and the sample median were determined by Laplace.[13] The distribution of the sample median from a population with a density function is asymptotically normal with mean and variance[14]

where is the median value of distribution and is the sample size. In practice this may be difficult to estimate as the density function is usually unknown.

These results have also been extended.[15] It is now known for the -th quantile that the distribution of the sample -th quantile is asymptotically normal around the -th quantile with variance equal to

where is the value of the distribution density at the -th quantile.

Estimation of variance from sample data

The value of —the asymptotic value of where is the population median—has been studied by several authors. The standard 'delete one' jackknife method produces inconsistent results.[16] An alternative—the 'delete k' method—where grows with the sample size has been shown to be asymptotically consistent.[17] This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,[18] but converges very slowly (order of ).[19] Other methods have been proposed but their behavior may differ between large and small samples.[20]

Efficiency

The efficiency of the sample median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size and on the underlying population distribution. For a sample of size from the normal distribution, the ratio is[21]

The coefficient of dispersion (CD) is defined as the ratio of the average absolute deviation from the median to the median of the data.[23] It is a statistical measure used by the states of Iowa, New York and South Dakota in estimating dues taxes.[24][25][26] In symbols

where n is the sample size, m is the sample median and x is a variate. The sum is taken over the whole sample.

Confidence intervals for a two sample test where the sample sizes are large have been derived by Bonett and Seier[23] This test assumes that both samples have the same median but differ in the dispersion around it. The confidence interval (CI) is bounded inferiorly by

where tj is the mean absolute deviation of the jth sample, var() is the variance and zα is the value from the normal distribution for the chosen value of α: for α = 0.05, zα = 1.96. The following formulae are used in the derivation of these confidence intervals

where r is the Pearson correlation coefficient between the squared deviation scores

and

a and b here are constants equal to 1 and 2, x is a variate and s is the standard deviation of the sample.

Previously, this article discussed the concept of a univariate median for a one-dimensional object (population, sample). When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one. In higher dimensions, however, there are several multivariate medians.[22]

The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.[22][27]

where X and a are vectors, if this expectation has a finite minimum; another definition is better suited for general probability-distributions.[10][22] The spatial median is unique when the data-set's dimension is two or more.[10][11][22] It is a robust and highly efficient estimator of a central tendency of a population.[28][22]

The Geometric median is the corresponding estimator based on the sample statistics of a finite set of points, rather than the population statistics. It is the point minimizing the arithmetic average of Euclidean distances to the given sample points, instead of the expectation. Note that the arithmetic average and sum are interchangeable since they differ by a fixed constant which does not alter the location of the minimum.

For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population pseudo-median, which is the median of a symmetrized distribution and which is close to the population median.[citation needed] The Hodges–Lehmann estimator has been generalized to multivariate distributions.[29]

This is a method of robust regression. The idea dates back to Wald in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent parameter : a left half with values less than the median and a right half with values greater than the median.[31] He suggested taking the means of the dependent and independent variables of the left and the right halves and estimating the slope of the line joining these two points. The line could then be adjusted to fit the majority of the points in the data set.

Nair and Shrivastava in 1942 suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples.[32] Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.[33] Tukey combined these ideas and recommended dividing the sample into three equal size subsamples and estimating the line based on the medians of the subsamples.[34]

An estimate of a one-dimensional parameter θ will be said to be median-unbiased if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.

—page 584

Further properties of median-unbiased estimators have been reported.[36][37][38][39] In particular, median-unbiased estimators exist in cases where mean-unbiased and maximum-likelihood estimators do not exist. Median-unbiased estimators are invariant under one-to-one transformations.

The idea of the median originated[citation needed] in Edward Wright's book on navigation (Certaine Errors in Navigation) in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations.

In 1774, Laplace suggested the median be used as the standard estimator of the value of a posterior pdf. The specific criteria was to minimize the expected magnitude of the error; |α - α*| where α* is the estimate and α is the true value. Laplaces's criterion was generally rejected for 150 years in favor of the least squares method of Gauss and Legendgre which minimizes < (α - α*)2 > to obtain the mean. [41] The distribution of both the sample mean and the sample median were determined by Laplace in the early 1800s.[13][42]

Antoine Augustin Cournot in 1843 was the first[citation needed] to use the term median (valeur médiane) for the value that divides a probability distribution into two equal halves. Gustav Theodor Fechner used the median (Centralwerth) in sociological and psychological phenomena.[43] It had earlier been used only in astronomy and related fields. Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace.[43]